# There is an infinite chessboard...

Dejection
Posers and Puzzles 10 Jun '08 02:33
1. 10 Jun '08 02:33
There is an infinite chess board, with a horizontal line across the whole length of it. Suppose you place as many pieces as you want below the line. A move consists of jumping a piece over another orthoganally, and a jump may only be made over exactly one piece, and there must be a vacant space to jump to. The piece that is jumped over is removed. Is it possible to reach the 5th row above the line in a finite number of moves?
2. 10 Jun '08 05:24
No... you can't put pieces above the line only below so you could only cross the line, right?
3. 10 Jun '08 07:001 edit
On a normal chessboard: suppose the line between the 4th and 5th ranks is our line.
Pieces on d4 and d3, d3-d5 first row
Pieces on d4, e4, f4, d4: d3-d5, f4-d4, d4-d6 second row
and so it continues.
You can certainly get it to the fourth row, but is the fifth possible?
4. 10 Jun '08 17:45
Hint: assign a numerical value to each square of the chessboard, in such a way that in every move (jump) the sum of the values of the occupied squares stays the same or decreases.
5. eldragonfly
leperchaun messiah
10 Jun '08 19:541 edit
Originally posted by Dejection
There is an infinite chess board, with a horizontal line across the whole length of it. Suppose you place as many pieces as you want below the line. A move consists of jumping a piece over another orthoganally, and a jump may only be made over exactly one piece, and there must be a vacant space to jump to. The piece that is jumped over is removed. Is it possible to reach the 5th row above the line in a finite number of moves?
No. The stipulation that "The piece that is jumped over is removed" would seem to indicate that no piece would be able to get beyond the first row beyond the line on the infinite chess board.
6. 10 Jun '08 20:063 edits
Originally posted by eldragonfly
The stipulation that "The piece that is jumped over is removed" would seem to indicate that no piece would be able to get beyond the first row beyond the line on the infinite chess board.
Uh, no it wouldn't.

Remember that jumps can go left and right too, not just up and down. Dejection's little example above shows explicitly how you can get beyond the first row beyond the line (although I think one of his d4 pieces initially listed should be a d3 in his example of getting to the second row beyond the line).
7. 10 Jun '08 20:282 edits
edit... just saw lemon's edit
8. eldragonfly
leperchaun messiah
10 Jun '08 20:461 edit
Originally posted by LemonJello
Uh, no it wouldn't.

Remember that jumps can go left and right too, not just up and down. Dejection's little example above shows explicitly how you can get beyond the first row beyond the line (although I think one of his d4 pieces initially listed should be a d3 in his example of getting to the second row beyond the line).
Well alright the second line then. good grief read Dejections post, he only shows that a piece can make it over the line to the 5th rank man! ðŸ˜› If pieces are removed then i don't see how this is possible. Please don't make me fen diagram this out for you lemmonjelly.
9. eldragonfly
leperchaun messiah
10 Jun '08 20:54
Originally posted by Dejection
On a normal chessboard: suppose the line between the 4th and 5th ranks is our line.
Pieces on d4 and d3, d3-d5 first row
Pieces on d4, e4, f4, d4: d3-d5, f4-d4, d4-d6 second row
and so it continues.
You can certainly get it to the fourth row, but is the fifth possible?
you forgot to remove the piece on d4 after the d3-d5 jump occurs.
10. 10 Jun '08 21:001 edit
Originally posted by eldragonfly
Well alright the second line then. good grief read Dejections post, he only shows that a piece can make it over the line to the 5th rank man! ðŸ˜› If pieces are removed then i don't see how this is possible. Please don't make me fen diagram this out for you lemmonjelly.
he only shows that a piece can make it over the line to the 5th rank man!

Although, as I said before, I think his post has a minor typo, he shows how even on a finite board you can get to the second row beyond the line (in his example, the 6th rank). He's giving you a hint about how to proceed. What do you want him to do? Give you the solution outright? Would you like for him to tuck you in bed, serve you warm milk, and sing you sweet lullabies every night too?
11. 10 Jun '08 21:012 edits
Originally posted by eldragonfly
you forgot to remove the piece on d4 after the d3-d5 jump occurs.
No he didn't. His jump from f4-d4 can only happen if d4 is empty.

So the removal of the piece on the d3-d5 jump is therefore implicit (it's also implicit in the first example he gives too).
12. 10 Jun '08 21:57
It's actually easily determined that you can't get over row five.

It's a variation of the "solitaire" problem if i recall correctly.
13. SwissGambit
Caninus Interruptus
10 Jun '08 22:18
Hmm. This seems too easy. I hope I'm understanding the stip correctly.

Let's say the line is between the 4th and 5th ranks.

1.d1-d5 2.f4-d4 3.d4-d6 4.f3-d3 5.d3-d7 6.f2-d2 7.d2-d8 8.f1-d1 9.d1-d9.
14. 10 Jun '08 23:16
Originally posted by SwissGambit
Hmm. This seems too easy. I hope I'm understanding the stip correctly.

[fen]8/8/8/8/3RRR2/4RR2/4RR2/3RRR2 w - - 0 1[/fen]

Let's say the line is between the 4th and 5th ranks.

1.d1-d5 2.f4-d4 3.d4-d6 4.f3-d3 5.d3-d7 6.f2-d2 7.d2-d8 8.f1-d1 9.d1-d9.
i thought you could only jump 1 square at a time
else if dejection managed to get to row 4, its seems easy that on an infinite board he would be able to get to row 5 as well as row 6 and so on
15. eldragonfly
leperchaun messiah
10 Jun '08 23:21
Originally posted by banx99
i thought you could only jump 1 square at a time
Same here i thought the same since it doesn't state explicity that empty square jumps are legal or not legal, and the example given seems to indicate only single square/jumps.